Let Fq be a finite field with q=pn elements. In this paper, we study the number of solutions equations form a1x1d1+…+asxsds=b xi?Fpti, where ai,b?Fq and ti|n for all i=1,…,s. our main results, employ results on quadratic forms to give an explicit formula diagonal restricted solution sets satisfying certain natural restrictions exponents. As consequence, present conditions existence solutions. s...

In this paper, a method for constructing near optimal normal basis algebraic extensions of finite field is described. each extension, except the squares elements, product two distinct elements can be expressed as linear combination those with coefficients in much smaller subfield.

The puncturing and shortening techniques are two important approaches to constructing new linear codes from old ones. In the past 70 years, a lot of progress on technique has been made, many works punctured have done. Many families with interesting parameters obtained technique. However, little research done there only handful references shortened codes. first objective this paper is prove some...

In this paper, we explore the existence of m-term arithmetic progressions in $${\mathbb {F}}_{q^n}$$ with a given common difference whose terms are all primitive elements, and at least one them is normal. We obtain asymptotic results for $$m \ge 4$$ concrete \in \{2,3\}$$ , where complete list exceptions when belongs to {F}}_{q}$$ obtained. The proofs combine character sums, sieve estimates com...